Textbook Chapter 1

Section 1.5 Functions and Logarithms 43
Table 1.15 Saudi Arabia’s
Natural Gas Production
Year Cubic Feet (trillions)
1997 1.60
1998 1.65
1999 1.63
2000 1.76
2001 1.90
Source: Statistical Abstract of the
United States, 2004–2005.
Solve Algebraically
(1.0525) t 2.5 Divide by 1000.
ln (1.0525) t ln 2.5
t ln 1.0525 ln 2.5 Power Rule
ln 2.
5
t 17.9
ln 1. 0525
Take logarithms of both sides.
Interpret The amount in Sarah’s account will be $2500 in about 17.9 years, or about
17 years and 11 months. Now try Exercise 47.
f (x) = 0.3730 + (0.611) ln x
X = 12
[–5, 15] by [–1, 3]
Y = 1.8915265
Figure 1.39 The value of f at x 12 is
about 1.89. (Example 7)
1
EXAMPLE 7
Estimating Natural Gas Production
Table 1.15 shows the annual number of cubic feet in trillions of natural gas produced by
Saudi Arabia for several years.
Find the natural logarithm regression equation for the data in Table 1.15 and use it to
estimate the number of cubic feet of natural gas produced by Saudi Arabia in 2002.
Compare with the actual amount of 2.00 trillion cubic feet in 2002.
SOLUTION
Model We let x 0 represent 1990, x 1 represent 1991, and so forth. We compute
the natural logarithm regression equation to be
f (x) 0.3730 (0.611) ln(x).
Solve Graphically Figure 1.39 shows the graph of f superimposed on the scatter
plot of the data. The year 2002 is represented by x 12. Reading from the graph we
find f (12) 1.89 trillion cubic feet.
Interpret The natural logarithmic model gives an underestimate of 0.11 trillion cubic
feet of the 2002 natural gas production. Now try Exercise 49.
Quick Review 1.5 (For help, go to Sections 1.2, 1.3, and 1.4.)
In Exercises 1–4, let f (x) 3 x, 1 g(x) x 2 1, and evaluate the
expression.
1. ( f g)(1) 1
2. (g f )(7) 5
3. ( f g)(x) x 2/3
4. (g f )(x) (x 1) 2/3 1
In Exercises 5 and 6, choose parametric equations and a parameter
interval to represent the function on the interval specified.
1
5. y , x 2 6. y x, x 3
x 1
Possible answer:
Possible answer:
1
x t, y , t 2
x t, y t, t 3
t 1
In Exercises 7–10, find the points of intersection of the two curves.
Round your answers to 2 decimal places.
7. y 2x 3, y 5 (4, 5)
8. y 3x 5, y 3 8 3 , 3 (2.67, 3)
9. (a) y 2 x , y 3 (1.58, 3)
(b) y 2 x ,
y 1 No intersection
10. (a) y e x , y 4 (1.39, 4)
(b) y e x ,
y 1 No intersection